Render Animation renders the same frame each time while Wiggle Bones addon is installed
HOW TO RENDER ANIMATION WITH WIGGLE ARMATURE:

Select all bones with Wiggle settings applied and bake them in
the wiggle bone properties area of the Bone tab. When you do this,
that panel becomes deactivated.
…however, rendering animation will still fail until you go to the
armature tab, scroll to the bottom, and uncheck “Wiggle Armature”.